The Geometric Origin of Spacetime: A Formal Rotor-Dynamic Route to the Einstein Field Equations

General Relativity describes gravity as the curvature of spacetime, while field theory describes matter and radiation as dynamical fields defined on that spacetime. This separation leaves unresolved the deeper question of why metric geometry and matter stress-energy should belong to a single equation. This paper develops a formal rotor-dynamic route to the Einstein Field Equations by treating spacetime geometry and matter as different long-wavelength sectors of a common four-dimensional rotor substrate. The scalar Rotor Curvature Field ψ, previously used to describe curvature amplitude and phase, is extended to an orientation-valued field Ψ capable of defining a local rotor frame eᵃ_μ. From this frame an effective Lorentzian metric is constructed by g_μν = ηₐb eᵃ_μeᵇ_ν. Variation of neighboring rotor frames defines a connection whose curvature maps to the Riemann tensor of the effective metric. In the long-wavelength limit, the generally covariant rotor action is dominated by the Einstein-Hilbert term, while higher-curvature corrections are suppressed by powers of r²/L². Localized rotor solitons form the matter sector, with stress-energy obtained by metric variation of the matter action. Stationary variation of the total effective action yields G_μν + Λg_μν = 8πG/c⁴ · T_μν. Within this interpretation, G measures inverse substrate curvature stiffness, while Λ represents residual large-scale curvature pressure associated with the substrate’s expansion geometry. The result frames General Relativity as the leading macroscopic self-consistency condition of a deeper curvature-circulation substrate.